Please give full and detailed solution for points. Thanks. a) In how many ways c

Question

Please give full and detailed solution for points. Thanks.

a) In how many ways can 15 pennies be distributed to 5 children so that each child gets at least one penny? b) If we drop the requirement that each child gets at least one penny, how many ways are there? c) Suppose instead of 15 identical pennies, we want to distribute 15 different presents to 5 children. How many ways are to do this? Do not assume each child must get one present. d) Now suppose we wish to distribute 15 presents to 5 children so that each child gets at least one. How many ways are there to do this?

Explanation / Answer

Solution:

Solution:

As we know that genral formula

The total number of ways in which ‘n’ identical can be distributed to ‘p’ persons so that each person receives at least one item is = n-1Cp-1

here n = 15 pennies and p =5

therefore

The total number of ways in which ‘15’ pennies can be distributed to 5 children so that each child receives at least one item

= 15-1C5-1

=14C4

= 1001 ways

(b)

Now each child gets any number either 0 or all

The total number of ways in which ‘15’ pennies can be distributed to 5 children so that each child receives any number of pennies

= 15+5-1C5-1

= 3876 ways

(c) In this case identical things become distinct objects

here n = 15 distinct object

and we know that

In general, the combination of n distinct objects taken r at a time, is represented and calculated as:

=nCr

here n = 15 and r = 5

Answer: 15C5 = 3003

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